One could argue that this is not the only possible choice for the
random perturbation. On one hand we chose a
multiplicative noise, instead of an additive one, but this is due to
technical reasons (see [1]}).
On the other hand, there are other possible choices, for example the
Brownian motion could depend on the father and
not on the node itself, so that brothers would share the same Brownian
motion. But the choice we made is dictated by the fact that we'd like to
have a formal conservation of the energy, as we have in the
deterministic case (see [2]}).
If we use Itô formula to calculate
\(\frac{1}{2} dX_j^2 &= X_j\circ dX_j \\&= (c_jX_{\bar{\jmath }}^2X_j-\sum _{k\in {\mathcal {O}}_j}c_kX_j^2X_k)dt + \sigma c_jX_{\bar{\jmath }}X_j\circ dW_j - \sigma \sum _{k\in {\mathcal {O}}_j}c_kX_jX_k\circ dW_k,\) 
